SLAC-PUB-2489 March 1980 (T/E) TARGET MASS CORRECTIONS IN QCD* W. R. Fraser? and J. F. Gunion 9 Stanford Linear Accelerator Center Stanford University, Stanford, California 94305 ABSTRACT We employ the diagrammatic approach to scale breaking, which allows for the fact that the target quark in a hadron is necessarily offshell, to demonstrate that Nachtmann moments do not generally absorb all gadron/q2 corrections. Submitted to Physical Review Letters * Work supported in part by the Department of Energy, contract DE-AC03-76SF00515, in part by the University of California at Davis and at San Diego, and in part by the A. P. Sloan Foundation. t Permanent address: University of California, San Diego, Physics Department, B-019, La Jolla, CA 92093. 3 Permanent address: University of California, Davis Physics Department, Davis, CA 95616.
-2- Experimental testing of asymptotically-free QCD requires a momentum transfer Q2 large enough to make an expansion in powers of as(q2) valid. Moreover, Q2 must be large compared to all relevant masses. At the energies at which most tests have been carried out, however, the ratio M2/Q2, where M is the mass of the target, is not neglibible. Such targetmass corrections greatly complicate the interpretation of deep inelastic scattering data at moderate energies. Georgi and Politzerl inferred from the operator product expansion that the use of a new scaling variable 5 instead of the Bjorken x would, to a good approximation, absorb all target mass corrections. In standard notation where the lepton momentum transfer is q, the target momentum is p, Q2 = -q2, and p2 = M2, the variable 5 is defined by 5= 2x 1 + h+ 4MLx2 / Q2,._ (1) where x = Q 2 /2p*q. In terms of moments of structure functions, the claim is that the use of Nachtmann moments2 incorporates target mass corrections. For example, in deep inelastic neutrino scattering the Nachtmann moment of the F3 structure function is My : ujl % F3 gn+' [1 + (n+l)j1+4m2x2 / Q2 ] (2) x ' There has been extensive discussion and some controversy.in the literature as to the validity of the claim that the Nachtmann moments incorporate all target mass corrections.3 The diagrammatic method for the analysis of QCD4 now provides a systematic approach in which this question can be answered, without making the approximation, used in some
-3- previous #approaches, that the target quark is on-shell. This paper gives a brief summary of our results, which are negative: we find target mass corrections of the form M2/Q2 which are not incorporated by the use of Nachtmann moments. The advantages and limitations of Nachtmann moments can already be seen from analysis of the simple class of diagrams shown in Fig. 1, where the deep inelastic probe strikes a quark, and where K represents the two-particle irreducible amplitude for finding this quark inside the target. Let us first illustrate the simplest features of the problem by considering a model in which all the particles, including the deep inelastic probe, are scalar. We write a spectral representation for K KC&p) = $- do o(o,k2,p2) (p-k)2-cs (3) The deep inelastic amplitude of Fig. 1 is then T(p,q) = J- d4k do p(o,k2,p2) (2~)~ k4(k- q)2 C(p-k)2- crl ' (4) We will calculate T in the Euclidean domain, which is algebraically equivalent to taking p2 and q2 > 0 and changing the sign of o, and expand T(p,q) = 2 Mn(p2,q2) n=o n C,(P l 3, (5) where the Cn - CA are Gegenbauer polynomials. If the Mn so defined are continued to the physical region one finds that they are the Nachtmann moments appropriate to the spinless problem, 1 Mn(p2,q2) = $ dx en+ ; 0 x2 ImT(p,q). (6)
-4- One can perform the angular integration in (4) by expanding the denominators6 1 = (p-k)2 + o m c n=o pk n+l c,c; l i;> (7) where ZU pk = 2 p + k2 + o - J(p2+k2+o)2-4p2k2 2pk (8) Using standard properties of the Cn one finds Mn(p2,q2) = Jdop(o,k2,p2) n+l - q2) ] (9) If the function p is highly convergent the second term in the bracket is suppressed and (continuing to p2=-m2, q2=q2) Mn + AnOf. (10) Q In contrast an ordinary x-moment, in this same model, would have a series 2 2 of M /Q corrections.7 In general, we can write a Nachtmann moment as Mn = fn (')(Q~)A~')(M~)+ fl1)(q2) $ Ai (M2)+... (11) Q (i> The question we are examining is whether the An are negligible for irl. One might hope, for example, that they are of order as(q2) or at least as(m2), relative to A A")(M2). 0 r, one might argue, following De Rujula et a1.,3 that the correction terms would be characterized by a scale X 2 2 /Q where X is characteristic of the inverse hadron size rather
-5- than its mass. In fact we have shown that these are the only possibilities in the scalar model with strongly convergent p. (i.21) Exactly how many of the An vanish depends on the explicit form of p. For fixed CT, and p of the power-law form i.i p(k2) = A 2, k2i-x (12) we find A(l)= Ac2)= n n 0 and that explicit M2/Q2 corrections begin with A:) # 0. The Nacht mann moments continue to absorb the leading M2/Q2 corrections. We shall show that this simplicity of the scalar model does not persist in models with more realistic spin assignments; explicit M2/Q2 corrections appear at all orders. Perhaps the cleanest case is that of a virtual photon target,8 where the lowest-order diagrams are shown in Fig. 2. We compute the Feynman integrals as before, and look for leading Rnq2 terms. Here we give the result only for the simplest Nachtmann moment of the O(4)-spin- zero amplitude (13) This amplitude has a Gegenbauer expansion and Nachtmann moments just the same as the scalar case given by (5) and (6). We find -- 2 n+l 1 1 * (14)
-6- Explicit p2/q2 corrections are present, and Nachtmann moments do not have simple scaling properties for this photon target case. Similar results are obtained for other terms in the QCD ladder series. Moreover, the higher order terms in the series cannot cancel the p2/q2 corrections in (14) because each term contains a different number of intermediate-state particles, and these contribute additively to the imaginary part. Thus for the photon target the additional structure introduced by spin and by the inevitable presence of the crossed graph (from which all the (p2/q21k, k 2 2, terms arise) results in target mass corrections which are not incorporated simply by using Nachtmann moments or S-scaling. One might still hope, however, that the trouble lies in the anomalous character of the photon target-the point-like part of the photon contains quarks with very high transverse momentum. However even for a hadronic target the spin structure and the presence of nonleading graphs (related to gauge invariance) turn out to be essential complications; the damping provided by the hadron wave function will not generally suppress the target mass dependence by replacing it with X/Q2 where X is a parameter characterizing the average value of k2 of the quark in the hadron. In particular, we have examined this question using two different models of hadrons, and find that target mass corrections persist if the hadronic wave function falls off like a power at high k2, as in QCD.' We shall illustrate here only the simplest model we have.considered, in which we calculate the amplitude F3 for neutrino scattering on a massive charged spin one-half bound state (a "proton") with one charged spin one-half constituent of zero mass and one uncharged spin zero constituent of mass 6, shown in Fig. 3. At each proton vertex we supply
-7- a factor'p(k2) to represent the suppression of high k2 by the proton wave r function. The result of a calculation similar to the ones already describzd islo (15) Again, one sees explicit p2/q2 terms, which originate from the spin structure. In general, the phenomenological importance of these terms is difficult to evaluate, since they depend on u and on the form of p(k2). We can, however, examine them for the form of p given in Eq. (12) which at least has the high-k2 behavior of QCD. Since the large-k2 region controls the large-n behavior we concentrate on that limit, where we find 2 2 $+2!&-nE-- Q2 3nL. Q2 1 (16) If M2 and u are comparable (as expected for stability of the "proton") then the terms of the form nm2/q2 and no/q2 can be very important phenomenologically.ll Abbott and Barnett12 have shown that such terms, with an appropriate coefficient, can account for much if not all of the observed non-scaling behavior usually attributed to QCD anomalous dimensions. We find the numerical coefficients of these terms to be model dependent. (In particular, non-leading graphs, related to gauge invariance, typically change the sign of the nm2/q2 and nu/q2 terms.) Nevertheless, it is interesting that they arise in such simple diagrams as Fig. 3.
-8- To summarize, we have found- rigorously for a photon target and in models for a hadron target-that explicit target mass corrections of the - form M2/Q2 arising from spin structure and non-leading diagrams are not taken into account simply by using Nachtmann moments or S-scaling. It may still turn out that Nachtmann moments or c-scaling provide improved fits to the data, but this is an experimental and phenomenological question. We conclude with a comment on a limit in which consideration of target mass corrections simplifies; namely, the limit p29q2 + O3 ; r = q2/p2 > 1 fixed. (173 In this limit the QCD log development collapses. For instance if we compare the zero-gluon and one-gluon terms of the standard axial gauge ladder series we have (omitting wave function renormalization factors inessential to the argument) in the limit (17) 2 0-gluon + 1-gluon = 1 + y, J q drnk2 P2 Rnk2 -l+yn% En 4 and we see that the one-gluon contribution is suppressed by a factor as (Q2) l Thus the one-loop diagrams we have been calculating dominate in this limit. In fact, it is only in this limit that the p2/q2 corrections can be more important than the higher-order as effects, 2 %>A. 4 Rx1 q2 (19) In the usual fixed p2,q2 + m limit the sense of the inequality is reversed and higher order a S effects should be more important than target mass corrections. It is even possible that the low-q2 region of deep inelastic data on a nucleon target is closer to a fixed q2/p2 regime than to the
-9- regime in'which the leading log series of QCD, appropriate to the asymptotic q2 += 03 limit, has a chance to develop. In that case, simple one-loop diagrams such as we have been calculating should describe the data, with the scaling violation resulting entirely from M2/Q2 corrections! ACKNOWLEDGMENTS We would like to thank S. Drell for his kind hospitality while the authors were visiting SLAC. This work was supported in part by the Department of Energy, contract DE-AC03-76SF00515, and in part by the University of California at Davis and at San Diego. (One of US (JFG) thanks the A. P. Sloan Foundation and the University of California at San Diego for support during part of the work.)
-lo- REFERENCES 1. H. Georgi and H. D. Politzer, Phys. Rev. m, 1829 (1976). 2. 0. Nachtmann, Nucl. Phys. m, 237 (1973); V. Baluni and E. Eichten, Phys. Rev. g, 3045 (1976); S. Wandzura, Nucl. Phys. B122, 412 (1977). 3. D. J. Gross, S. B. Treiman and F. A. Wilczek, Phys. Rev. D&, 2486 (1977); A. De Rujula, H. Georgi and H. D. Politzer, Phys. Rev. a, 2495 (1977); R. Ellis, G. Parisi and R. Petronzio, Phys. Lett. m, 97 (1976); R. Barbieri, J. Ellis, M. K. Gaillard and G. Ross, Nucl. Phys. B117, 50 (1976); K. Bitar, P. Johnson and W.-K. Tung, Phys. Lett. m, 114 (1979). 4. See the review by A. Buras, Rev. Mod. Phys. 52, 199 (1980) and references therein. 5. For a more detailed account see W. Frazer, J. Gunion and J. Rodriquez-Espinosa, SLAC-PUB-in preparation. 6. M. Levine and R. Roskies, Phys. Rev. m, 421 (1974). 7. When one generalizes from the diagram of Fig. 1 to the ladder diagrams which g?ve rise to anomalous dimensions,4 then (10) is altered to Mn(.M2,Q2)w (1/Q2)(anQ2/n2)-YnAn(M2) but no new M2/Q2 correction terms develop. 8. See W. Frazer and J. Gunion, Phys. Rev. z, 147 (1979) and references therein. 9. S. Brodsky and P. Lepage, Phys. Lett. 43, 545 (1979).
-ll- 10. The'diagram in Fig. 3 gives rise to a gauge-invariant calculation of F even without including other diagrams in which the hadronic e3 vertex is probed. Such diagrams do, however, add to the corrections given in (17). See Ref. 5. 11. In the limit CT << M2 the x dependence of the structure function is highly unphysical. In any case a reanalysis of (15) in this limit yields multiple M2/Q2 corrections to My for n 2 3 (see Ref. 5). 12. L. F. Abbott and R. M. Barnett, SLAC-PUB-2227, Nov. 1979, submitted to Annals of Physics. FIGURE CAPTIONS 1. General "handbag" diagram for deep inelastic scattering on a hadron target. 2. Diagrams for deep inelastic scattering on a photon target. Both are required for gauge invariance. 3. Deep inelastic scattering on a spin-l/2 "proton" composed of a charged spin-l/2 massless "quark" and a charged spin-zero constituent of mass u.
- m cl w I m 0 2-! tq cd -. ca. h) n -. (D. I)